We are motivated by the Fisher model of population genetics. In this model, \(u(x,t)\) represents the relative density of one (say, \(a\)) of two gene alleles (\(a\) and \(A\)) at a particular gene locus in the population of a migrating diploid species. We consider the problem in a slightly more general setting: \[ \partial u/\partial t= \Delta u+ k\cdot\nabla u+[(f_1+ f_2)u- f_2]u(1- u), \quad (x,t)\in \mathbb{R}^n\times (0,\infty), \tag{1} \]
\[ u(x,0)= u_0 (x), \quad 0< u_0< 1, \quad \text{for }u\in \mathbb{R}^n. \] One might expect the solution \(u(x,t)\in (0,1)\) would asymptotically converge to 0, 1 or some limiting function of the form \(\widetilde {u} (\nu\cdot x-ct, t)\) as \(t\to \infty\). If \(u\equiv 0\), and \(u\equiv 1\) are linearly unstable, then all solutions would approach some propagating cline \(\widetilde {u} (\nu\cdot x-ct, t)\) which is periodic in the \(t\) variable. The term “propagating cline” is adopted from T. Nagylaki, Genetics 80, 595-615 (1975), in the expression of its propagation pattern. We give conditions on \(f_j\) for the existence of a cline solution which is an interesting pattern in genetics. Further we show that under the same conditions, the cline is unique, and it attracts any solution of (1) except for 0 and 1.